Vice President Joseph Biden and Sen. Mitch McConnell were locked in urgent talks late Sunday over the “fiscal cliff” after Democrats offered several significant concessions on taxes, including a proposal to raise rates only on earnings over $450,000 a year.

With a New Year’s Eve deadline hours away, Democrats abandoned their earlier demand to raise tax rates on household income over $250,000 a year, as President Obama vowed during the recent presidential campaign.

They also relented on the politically sensitive issue of the estate tax, according to a detailed account of the Democratic offer obtained by The Post, promising to stage a vote in the Senate that would guarantee that taxes on inherited estates remain at their current low levels, a key GOP demand.

Still, McConnell (R-Ky.) was holding out to set the income threshold for tax increases even higher, at $550,000, according to people close to the talks in both parties. And he was protesting a Democratic proposal to raise taxes on investment profits for households with income above $250,000.